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Chapter 4 | Applications of Derivatives
Figure 4.66 How can we choose x and y to minimize the travel time from the cabin to the island?
Step 2: The problem is to minimize T . Step 3: To find the time spent traveling from the cabin to the island, add the time spent running and the time spent swimming. Since Distance = Rate × Time ( D = R × T ), the time spent running is T running = = x
D running R running
8 ,
and the time spent swimming is
D swimming R swimming
y 3 .
T swimming =
=
Therefore, the total time spent traveling is
y 3 .
T = x
8 +
Step 4: From Figure 4.66 , the line segment of y miles forms the hypotenuse of a right triangle with legs of length 2mi and 6− x mi. Therefore, by the Pythagorean theorem, 2 2 +(6− x ) 2 = y 2 , and we obtain y = (6− x ) 2 +4. Thus, the total time spent traveling is given by the function T ( x ) = x 8 + (6− x ) 2 +4 3 . Step 5: From Figure 4.66 , we see that 0≤ x ≤6. Therefore, ⎡ ⎣ 0, 6 ⎤ ⎦ is the domain of consideration. Step 6: Since T ( x ) is a continuous function over a closed, bounded interval, it has a maximum and a minimum. Let’s begin by looking for any critical points of T over the interval ⎡ ⎣ 0, 6 ⎤ ⎦ . The derivative is
−1/2
⎡ ⎣ (6− x ) 2 +4 ⎤ ⎦
(6− x ) 3 (6− x ) 2 +4 .
1 2
T ′( x ) = 1
x ) = 1
·2(6−
8 −
3
8 −
If T ′( x ) =0, then
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